Problems of Information Transmission

, Volume 53, Issue 3, pp 251–259 | Cite as

Propelinear codes related to some classes of optimal codes

  • I. Yu. Mogilnykh
  • F. I. Solov’eva
Coding Theory


A code is said to be propelinear if its automorphism group contains a subgroup that acts regularly on codewords. We show propelinearity of complements of cyclic codes C 1,i , (i, 2 m − 1) = 1, of length n = 2 m − 1, including the primitive two-error-correcting BCH code, to the Hamming code; the Preparata code to the Hamming code; the Goethals code to the Preparata code; and the Z4-linear Preparata code to the Z4-linear perfect code.


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  1. 1.
    Rifà, J., Basart, J.M., and Huguet, L., On Completely Regular Propelinear Codes, Proc. 6th Int. Conf. on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC-6), Rome, Italy, July 4–8, 1988. Mora T., Ed., Lect. Notes Comp. Sci., vol. 357. Berlin: Springer, 1989, pp. 341–355.Google Scholar
  2. 2.
    Borges, J., Mogilnykh, I.Yu., Rifà, J., and Solov’eva, F.I., Structural Properties of Binary Propelinear Codes, Adv. Math. Commun., 2012, vol. 6, no. 3, pp. 329–346.CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Mogil’nykh, I.Yu., On Extending Propelinear Structures of the Nordstrom-Robinson Code to the Hamming Code, Probl. Peredachi Inf., 2016, vol. 52, no. 3, pp. 97–107 [Probl. Inf. Trans. (Engl. Transl.), 2016, vol. 52, no. 3, pp. 289–298].MATHMathSciNetGoogle Scholar
  4. 4.
    Hammons, A.R., Jr., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., and Solé, P., The Z4-Linearity of Kerdock, Preparata, Goethals, and Related Codes, IEEE Trans. Inform. Theory, 1994, vol. 40, no. 2, pp. 301–319.CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Preparata, F.P., A Class of Optimum Nonlinear Double-Error-Correcting Codes, Inform. Control, 1968, vol. 13, no. 4, pp. 378–400.CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Dumer, I.I., Some New Uniformly Packed Codes, in Proc. Moscow Inst. Physics and Technology, Moscow, 1976, pp. 72–78.Google Scholar
  7. 7.
    Baker, R.D., van Lint, J.H., and Wilson, R.M., On the Preparata and Goethals Codes, IEEE Trans. Inform. Theory, 1983, vol. 29, no. 3, pp. 342–345.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Borges, J., Phelps, K.T., Rifà, J., and Zinoviev, V.A., On Z4-linear Preparata-like and Kerdock-like Codes, IEEE Trans. Inform. Theory, 2003, vol. 49, no. 11, pp. 2834–2843.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Rifà, J. and Pujol, J., Translation-Invariant Propelinear Codes, IEEE Trans. Inform. Theory, 1997, vol. 43, no. 2, pp. 590–598.CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Zinoviev, D.V. and Zinoviev, V.A., On the Preparata-like Codes, in Proc. 14th Int. Workshop on Algebraic and Combinatorial Coding Theory (ACCT-14), Svetlogorsk, Russia, Sept. 7–13, 2014, pp. 342–347.Google Scholar
  11. 11.
    Mogilnykh, I.Yu. and Solov’eva, F.I., Transitive Nonpropelinear Perfect Codes, Discrete Math., 2015, vol. 338, no. 3, pp. 174–182.CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Kasami, T., Lin, S., and Peterson W.W., Some Results on Cyclic Codes Which Are Invariant under the Affine Group and Their Applications, Inform. Control, 1967, vol. 11, pp. 475–496.CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Melas, C.M., A Cyclic Code for Double Error Correction, IBM J. Res. Develop., 1960, vol. 4, no. 3, pp. 364–366.CrossRefMATHGoogle Scholar
  14. 14.
    MacWilliams, F.J. and Sloane, N.J.A., The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. Translated under the title Teoriya kodov, ispravlyayushchikh oshibki, Moscow: Svyaz’, 1979.MATHGoogle Scholar
  15. 15.
    Semakov, N.V., Zinoviev, V.A., and Zaitsev, G.V., Interrelation of Preparata and Hamming Codes and Extension of Hamming Codes to New Double-Error-Correcting Codes, Proc. 2nd Int. Sympos. on Information Theory, Tsakhkadzor, Armenia, USSR, Sept. 2–8, 1971. Petrov, P.N. and Csaki, F., Eds., Budapest: Akad. Kiadó, 1973, pp. 257–263.Google Scholar
  16. 16.
    Semakov, N.V. and Zinoviev, V.A., Complete and Quasi-complete Balanced Codes, Probl. Peredachi Inf., 1969, vol. 5, no. 2, pp. 14–18 [Probl. Inf. Trans. (Engl. Transl.), 1969, vol. 5, no. 2, pp. 11–13].MATHMathSciNetGoogle Scholar
  17. 17.
    Kantor, W.M., On the Inequivalence of Generalized Preparata Codes, IEEE Trans. Inform. Theory, 1983, vol. 29, no. 3, pp. 345–348.CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Wan, Z.-X., Quaternary Codes, Singapore: World Sci., 1997.CrossRefMATHGoogle Scholar

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© Pleiades Publishing, Inc. 2017

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsBranch of the Russian Academy of SciencesNovosibirskRussia

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